Mechanical amplitude transformer



March 30, 1965 E. EISNER 3,175,406

MECHANICAL AMPLITUDE TRANSFORMER Filed March 25, 1963 6 Sheets-Sheet 1 0 I I I I l O FIG. 2

M 7 l |.o

o I I l O |.o -o.5 0.5 no

INVENTOR E. E/SNER BY ATTORNEY March 30, 1965 Filed March 25, 1963 FLEXURAL COMPLIANCE, Co

E. EISNER MECHANICAL AMPLITUDE TRANSFORMER 6 Sheets-Sheet 2 FIG. 3

P. M=2OO o m =o(l-.e., THIRD-ORDER FOURIER) 4 SELECTED OPTIMUM FORM March 30, 1965 E. EISNER MECHANICAL AMPLITUDE TRANSFORMER Filed March 25, I963 6 Sheets-Sheet 4 FIG. 5A

A o- PROFILE DISPLACEMENT (APPROXIMATE IDEAI.)

(a) STEPPED VIBRATOR FILLT RADIUS: MINOR DIAMETER FIG. 5B

b) FouRIER" VIBRATOR i MAx 0 U 20 I0- DISPLACEMENT ,42

40- STRAIN (zu /A) 20- 44 o l l l March 30, 1965 E. EISNER 3,175,

MECHANICAL AMPLITUDE TRANSFORMER Filed March '25, 1963 e Sheets-Sheet 5 M=2O (c) EXPONENTIAL VIBRATOR p=2. I632! .L AMAXO 2o 52 gm DISPLACEMENT V u STRAIN o I l I FIG. 6

March 30, 1965 E EISNER 3,175,406

MECHANICAL AMPLITUDE TRANSFORMER Filed March 25, 1963 6 Sheets-Sheet 6 FIG. 9

United States Patent 3,175,406 MECHANICAL AMPLITUDE TRANSFORMER Edward Eisner, Bernardsville, Nail, assignor to Bell Teleiphone Laboratories, Incorporated, New York, N.Y., a corporation of New York Filed Mar. 25,1963, Ser. No. 267,718 5 Claims. (Cl. 74-1) This invention relates to elongated mechanical vibratory members for amplifying the magnitude of repetitive physical displacements.

Reference may be had to United States Patent No. 2,573,168, granted October 30, 1951, to W. P. Mason and R. F. Wick, and to W. P. Masons book entitled Physical Acoustics and the Properties of Solids, published by D. Van Nostrand and Co., Princeton, New Jersey, 1958, Chapter VI, pages 156-478, for information relating to elongated mechanical vibratory members exponentially tapered to effect an amplification of longitudinally vibratory motion.

It is also known in the art that a stepped vibratory member having two sections of similar transverse crosssectional shape but radically differing cross-sectional areas, each section being substantially one-quarter wavelength long, can be employed to eifect an amplification of longitudinally vibratory motion. Stepped members are discussed, for example, by L. Balamuth in The Transactions of the Institute of Radio Engineers, Profesisonal Group on Ultrasonics Engineering (2), November 1954; see particularly section 8 on page 29.

All such members as are of interest in connection with the present invention have a maximum transverse dimension which is a fraction of a wavelength, for example, less than one-half and preferably less than one-quarter wavelength of the energy being amplified. Such members are further in the order of one-half to three-quarters of a wavelength long.

For reasons which will become more readily apparent from a perusal of the descriptive material given hereinunder, the stepped type of vibratory member mentioned above is limited as a practical matter by the strains generated in the member to relatively much smaller amplitude ratios or magnifications of vibratory amplitude than those which are frequently desirable. As used for comparative purposes throughout this specification, the ideal characteristics of the stepped type member are indicated. Accordingly, the Fourier type will, in actual practice, he better as compared with an actual stepped type than indicated by the present application.

Chapter VI of Masons book, mentioned above, discusses numerous uses for which devices of the invention and Masons exponentially tapered members are both readily adapted.

While Masons exponentially tapered members and the related conically tapered members, and the like, can, as

a practical matter, provide much higher amplitude ratios or magnifications of vibratory amplitude than the stepped members, the exponentially tapered and similarly shaped members, for large amplitude ratios, become so slender over a considerable fraction of their total length that they may be difficult to manufacture and difiicult to use since unwanted fiexural vibration or motion of the slender portion of the member can scarcely be avoided. Stated in other words, they lack sufiicient stiffness at large amplitude ratios to function in a completely satisfactory manner.

Accordingly, in accordance with the principles of the present invention, a new class of elongated vibratory members is proposed. The members of this new class are capable of providing very large magnifications of the vibratory energy and at the same time have a degree of Bil/5AM Patented Mar. 30, 1965- stiffness which is entirely sufiicient for the contemplated uses of the members. Since, as will presently appear, the longitudinal shaping of members of this new class is determined by the employment of Fourier series of third and fourth orders, the members are referred to throughout this application as Fourier vibrators.

A principal object of the present invention therefore is to eliminate difficulties encountered with prior art types of elongated mechanical vibratory members for amplifying the magnitude of repetitive physical displacements.

A further principal object is to provide vibratory members with greater ability to withstand large amplitudes.

A still further object is to provide vibratory members of enhanced stiffness at large values of magnification.

Other and further objects, features and advantages of the invention will become apparent from a perusal of the following detailed description of illustrative devices and from the accompanying drawing, in which:

FIGS. 1 and 2 illustrate the variation of the ratio of maximum to minimum transverse areas of a member of the invention for a specific magnification with several design parameters of the members;

FIG. 3 illustrates the effect upon fiexural compliance of variations in the ratio of maximum particle velocity to maximum permissible strain for six values of the magnification M of a member of the invention;

FIG. 4 presents comparative data relative to the magnification, fiexural compliance and ratio of maximum particle velocity to maximum permissible strain for vibrators of the stepped, exponential and Fourier types;

FIGS. 5A, 5B and 5C present comparative profiles and displacement and strain characteristics for corresponding vibrators of the stepped, Fourier and exponential types, respectively, for a magnitfication M of 20;

FIG. 6 presents comparative profiles of members of the stepped, exponential and Fourier types for a magnification M of FIG. 7 presents in graphic form the comparative relations of magnification M to the ratio of maximum to minimum diameters for members of the stepped, exponentia and Fourier types;

FIG. 8 is a perspective showing providing a direct comparison between a member of the invention and an exponentially tapered member of the above-mentioned patent to Mason et al., both members providing a magnification of ten; and

FIG. 9 is a perspective showing of a member of the invention providing a magnification of 100.

It can be shown that in most acoustically resonant solid bodies the maximum particle velocity, v,,,, is related to the speed of sound, 0, in the material, and to the maximum permissible strain S in the body by:

where (p is, in general, a (nondimensional) function of the geometry and of Poissons ratio of the material.

We shall assume restrictions on the type of resonant system, sufiicient for (l) to be true; namely, that the strains at any two points in the system be proportional to each other and that the speed of sound in the material be independent of amplitude of displacement or of strain. These conditions are necessary for most analyses of wave propagation, and do not seriously restrict the utility of Equation 1.

The dependence of (p on Poissons ratio expresses the fact that, in general, a single speed of sound does not describe the wave propagation, and a single component does not describe the strain. However, in many practical cases, including those of interest in the present application, either the stress or the strain can be described in terms of a single component and in that case go is a function of geometry only. Indeed, in many important cases the dependence of (p on shape is also restricted, for instance, if the sound propagates as plane, unidirectional waves, then the ratio of dimensions parallel and perpendicular to the propagation direction does not affect (p as long as the lateral dimensions are much smaller than the wavelength, for example, less than one-quarter wavelength.

Further, for bodies, such as those of interest in connection with the present invention, that do not contain erious stress concentrators, and in which most of the mass is not in unstressed, passenger material, (,0 does not in many instances differ greatly from 1. For example, for a uniform bar in longitudinal resonance, :1; for a uniform cantilever of circular section, =0.50; for a uniform tube (including a rod) in torsion, =1; for an exponential horn such for example as is disclosed in United States Patent No. 2,573,168, granted October 30, 1951, to W. P. Mason and R. F. Wick, with a 10: l ratio of end diameters, =2.0. Even for special shapes designed specifically for large (p, a value as large as 5 can be attained only by very careful design.

Thus, not only is an excellent figure of merit for assessing the ability of a particular body to resonate with large amplitude without overstraining, but Equation 1 also provides a ready means of deciding how easily a particular particle velocity can be obtained. If v,,,/ (cS is less than or equal to 03, almost any sensible design will do; if 0.3 is less than v /(cS which is less than or equal to 2.0, there is considerable choice among specially designed, tapered shapes; if 2.0 is less than v /(cS which is less than or equal to 5.0 only the most carefully designed shapes will do; and if v /(cS is greater than 5.0 it is very improbable that any suitable shape can be found.

Consider the longitudinal resonance of a slowly tapering rod whose lateral dimensions are small compared with the wavelength, A, for extensional Waves; the differential equation for the amplitude of longitudinal displacement, u(x), at coordinate x is ln A=log, A (i.c., the natural logarithm of A) du d u where A(x) is the cross-sectional area.

The conventional way of tackling such a problem is to express A as a function of x and to attempt the solution of the resulting differential equation for u, given the boundary conditions. Algebraic solution is possible in only a few cases, and numerical solution is cumbersome. Furthermore, the object is usually to design a body that will resonate so that the displacement and its derivatives satisfy given conditions. There is no clear way of guessing the function that will do this.

By reversing the procedure and selecting a wave-function u(x), the function A(x) can be unambiguously deduced from Equation 2. If we devise a function u(x) that satisfies the boundary conditions of the problem and gives the values of displacement, strain, et cetera, that we require at specific points, then we get a differential equation for A from Equation 2. A(x) can be obtained by simple integration; and, by judicious choice of the form of u(x), this integration can often by done algebraically in terms of known functions.

in free resonance, has a pair of strain-free surfaces (ends) at one of which the amplitude is much larger than at the other. The body can then be driven at the lowamplitude surface (end) by a resonating transducer, as shown for example in the above-mentioned patent to Mason et al., to give magnified vibration at the highamplitude surface. Two well known types exist for the magnification of longitudinal vibration, namely, the exponential (see Mason et al. patent) and the stepped (described in detail, for example, by L. Balamuth, as noted hereinabovc).

By considering the conservation of momentum, it can be seen that in any useful amplitude transformer, the cross-sectional area near the low-amplitude end must be much bigger than that near the high-amplitude end. We shall call the ratio of the maximum to the minimum cross-sectional areas R and the corresponding ratio of diameters for members of circular cross section, R Thus mnx 12 min min) RA/ where D indicates diameter and A indicates area. If Equation 2 is to apply with good accuracy to a longitudinal vibrator, the maximum cross-sectional dimension must be small compared to a wavelength, say, preferably less than x/ 4, although approximate results, that is, satisfactory where optimum performance is not required, can be obtained for values up to one-half wavelength. The length of the vibrator will be of the order of M2, for example, in some instances it may be as great as threequarters of a wavelength. R therefore, for members of circular cross section (and R for members of square or other cross section) determines the slenderness of the transformer. Whether the transformer is used to drive another vibratory member, or is used directly as an ultrasonic drill or other tool, flexural rigidity is obviously desirable. Indeed, if the fiexural rigidity is too low, the transformer will not only be difficult to make accurately but will also be difficult to use since unwanted flexural vibrations are most likely to occur. Thus, too large a value of R (or of R is undesirable.

An exponentially tapered vibrator of amplitude magnification M has the diametral ratio (R =M. If M is large this gives a very slender rod. Values of M up to 200 or more may, for example, be needed. The stepped vibrator, on the other hand, has the diametral ratio (R /M, and this clearly gives a stiff shape. Unfortunately, the highest value of (p obtainable with the stepped vibrator is about 0.8. This is not only rather low for the majority of purposes, but, in particular, makes it entirely unsuitable when the higher amplitude ratios or magnifications are required, since stresses sufiicient to cause damage to the member may be generated.

FOURIER TYPE AMPLITUDE TRANSFORMERS Equation 2 can be rewritten in nondimensional (that and l is the length of the vibrator.

We shall now look for a function U(X) that satisfies boundary conditions that express the required character-v istics of our vibrator.

As boundary conditions, We have (1) The two ends of the vibrator are to be stress-free and therefore strain-free; thus, at the low-amplitude end U'(0)=0 and at the high amplitude end (2) The ratio of the amplitudes of vibration at the two ends is to be M, the negative sign expressing opposed phases. Thus,

where U is the value of U at the larger or lower amplitude end of the vibratory member.

(3) The gradient of the lateral surface (or slope of the longitudinal cross-sectional contour of the vibratory member) is to be finite everywhere.

We may rewrite Equation 6 as uog. A) (6 We must then ensure that, wherever U approaches zero, (U+Q U) approaches zero at least as rapidly. We therefore choose a function for U'(X) that has no zeros in the range zero is less than or equal to X which is less than or equal to 1, except for the essential, first order, zeros at X=0 and X=l. Then, two necessary conditions are:

U"(0)=l U o A design giving a particularly low value of R results if the surface gradient at the low-amplitude end is not merely restricted to being finite, but is set to zero. Now, since both U" and (U+Q U) are zero at X :0, we can see from Taylors Theorem that U"(0) is finite, and, if the gradient of the surface is to vanish,

U"'(0)=0 e) This immediately settles the difficulty that (8e) is not a U0 MQOUO s m If U(X) is expressed as polynomial, it has to be at least seventh order to satisfy the eight conditions (8a-8h). As this makes the problem complicated, we take instead The Fourier series of Equation 11 above is employed in accordance with the present invention in truncated form, that is, its expansion is carried at most to the fourth order (that is, N:4) though, as will presently appear, for some purposes the tlu'rd order (that is, N=3) will result in a good vibrating member. The fourth order is, however, preferable since it can provide a still smaller area ratio R (or for a vibrator of circular cross section a small diametral ratio R which, as taught in the present application, is particularly advantageous since the vibratory member will then have greater stiffness (lower flexural compliance). Expanded to the fourth order (that is, N=4) Equation 11 will, of course, take the form The virtue of this form is that conditions (8a), (8b), (8g) and (8h) are identically satisfied, leaving only four conditions to which to fit the values of u To do this, N=3 is sufficient, and With this form for U(X) the integration (60) can be carried out algebraically. Using the notation and and computing U'(X) and A(X) for a given M and y, we can find g0 and R (the ratio of maximum to minimum cross-sectional areas). We can thus plot ('y) and R M) for any given M, and this is done for M=2O in FIG. 1, curves 10 and 12 respectively. The graphs for other values of M (as long as M does not closely approach 1) are essentially similar. The important feature is that R W) has a sharp minimum (as illustrated by curve 12 of FIG. 1) and that ('y) changes slowly in the neighborhood of the corresponding value of curve 10, FIG. 1.

Since the flexural compliance is approximately proportional to R there is nothing to be gained by using any value of 7 other than that corresponding to minimum R Thus, while in principle the third order Fourier form of U(X), using N :3 in Equation 11, gives us a choice, for given M, of a variety of combinations of and R one of these combinations has such overwhelming advantage over the others that we have really found a single, useful shape for each M. While it will later be seen that this shape is an interesting one, the design has turned out to be essentially a single parameter type, like the stepped and exponential vibrators.

A two-parameter design, in which both M and (p or M and R are at choice can be obtained with the Fourier series of Equation 11, using a value of N24 and 01 the coefficient of cos 41rX, being left as a free parameter. Defining an arbitrary constant ca as the following expressions are obtained for the coefiicients of the terms of Equation 11 This determines U(X) and its derivatives, and it follows that:

The denominator of relation (14) is a cubic in cos 1rX and could therefore be factorized to allow algebraic integration for (In A). But the resulting expressions would be so complicated that nothing would be gained over numerical integration.

A program was therefore written for an International Business Machines Model 7090 computer to compute U(X) and A(X)/A and hence to and R for given values of M, 42 and 'y. This showed that, for all M and 0: in the ranges giving practicable values, R W) had a sharp minimum, as had already been found for m =0 (that is, for the third order Fourier form, discussed above). The program was therefore modified to enable it to find the minimum value, (R of R W),

either for all given values of or for 'y FIG. 2 shows (R fl curve 14, and curve 16, as functions of ca for M :20. For the range of M investigated, 5 to 200, the curves are essentially similar. The important feature is that, in the approximate range 0.5o +0.5, (RU changes rather little, and outside that range it increases rapidly. If the fiexural stiffness of the vibrator were a function of cross section only, there would be little to choose among values of 0: in this range. But this is true only if the length (which is proportional to 7) changes little, and FIG. 2 shows that this is not the case here. Therefore a subroutine was added to the programme of the computer to compute the static fiexural compliances of these nominally half wave vibrators, for given loading conditions. Compliance is, of course, the inverse of stifiness.

In FIG. 3, curves through 25, inclusive, the values of flexural compliance C and (p corresponding to m for given 04 are plotted against each other for several values of M as marked on each curve.

In general, it may be assumed that there is no point in using a design with reduced stiffness unless thereby a higher value of lp is gained, that is, that only those parts of the curves 20 through 25, inclusive, on which (8C /6 0 are important. Now, from FIG. 3, it is apparent that, except for the lowest values of M (which are of little interest), (8C /8 0 for only very small ranges of C or (p. Therefore we have gained very little freedom of independent choice of M and (p by our addition of a free parameter to U(X). In fact, for each value of M there is a value of 00 that gives nearly the greatest bending stiffness and nearly the greatest (p. However, it is also clear that this value is not oc =0, but a -0.3. While the difierence between these two configurations is not great for M 50, for large values of M the difference may be considerable. Thus, for M :200, the shape for a =-O.34 is twice as stifli as that for (1. :0, and this ratio continues to increase with increasing M.

From the curves of FIG. 3 values of 0: that give good compromises between stiifness and (p can be selected. The points corresponding to the third order Fourier series (that is, N=3) are indicated by small circles and the selected optimum fourth order Fourier series (N=4) points are indicated by the small arrows.

The numerical data computed for the four higher values of M for the corresponding vibrators represented by curves 22 through 25 are shown in Table I.

8 Table I PROFILES 0F FOURTH-ORDER FOURIER TRANSFORMERS M: Magnification M=20.000 M=50.000 Alpha42= 0.30000 Alpha42= 0.30000 Gamma 1.40950 Gamrna= 1.43210 P 1.61755 Phi= 1.60161 Z D/DO Z D/DO Phi =1.50287 Phi: 1.58041 Z D/DO Z D/DU In FIG. 4 the values of compliance, that is, the inverse of stiffness, for the selected cases (that is, various values of M) are plotted against (p, curve 28, together with the corresponding curves 27 and 29 for the stepped and the exponential types of mechanical transformers, or vibrators, respectively.

In FIGS. 5A, 5B and 50, the profiles 30, 40 and 50 of the stepped, the Fourier and the exponential amplitude transformers (or vibrators) with M=20 are shown, respectively, together with their displacement and strain functions, as illustrated by curves 32, 42 and 52 and 34, 44 and 54, respectively. The cross-sectional shapes of the three types of transformers are similar, and the maximum cross-sectional dimension and the wavelength, A, are the same for all the transformers (or vibrators). The trends in flexural stiffness (inverse of compliance) and in maximum strain are clearly apparent.

In FIG. 6 the clear advantages of the Fourier transformer or vibrator at a high magnification (M of 100) is apparent from an inspection of the profiles 60, and of the stepped, Fourier and exponential types, respectively.

FIG. 7 compares the ratios of maximum to minimum diameters R /R needed to produce given magnifications M with the exponential, curve 88; the Fourier, curve 90; and the stepped, curve 92, amplitude transformers. This shows that for the Fourier transformer the ratio for a given magnification is only about sixty percent greater than for the stepped. This comparison is particularly favorable to the Fourier transformer because the stepped transformer here considered is an idealized, one-dimensional vibrator. The magnification produced in practice by a stepped transformer of given R is probably very considerably smaller than that shown in FIG. 7, particularly at high magnification.

FIG. 8 is substantially an exact reproduction of the perspective showing of FIG. 1 of the above-mentioned patent to Mason et al. xecpet that member 12 is tapered in accordance with a truncated Fourier series as taught in the present application instead of being exponentially tapered as taught in the Mason et al. patent. Both members, that is, member 12 of the Mason et al. patent and member 12 of FIG. 8, are proportioned to provide the very modest magnification of ten. All other parts of FIG. 8 are intended to be identical with and as described in the Mason et al. patent for the corresponding designation numbers (not primed in the patent drawing and description). Asuming as a fair representative example, that transducer or motor element 1' of FIG. 8 and its corresponding element 1 of FIG. 1 of the patent both provide a particle velocity of 50 centimeters per second to the larger end of the tapered member, since both members have the same magnification ten, the smaller ends of both members Will have particle velocities of 500 centimeters per second.

It is obvious from inspection and comparison of the two figures (FIG. 8 of this application and FIG. 1 of the patent) that the Fourier tapered member 12. of the present application is much stiffer mechanically than the exponentially tapered member of the patent.

In FIG. 9, member 12" is a perspecitve showing of the member whose profile is designated 70 in FIG. 6 of this application. This member has a truncated Fourier taper and is proportioned to provide a magnification of 100. Assuming again that the transducer driving the larger end of the member imparts a particle velocity of 50 for each value of magnification, M. To examine the possibility of providing a useful choice of stiffness versus 90, and of finding a design with even higher stiffness, a fourth-order Fourier series or displacement function is analyzed, Equation 11, N :4, with the following results. For M 10, it possible to achieve a useful increase in stiffness by sacrificing (p for l0 M 40 the stiffness varies very little over a wide range of go, and the thirdorder form, which is shown by the points indicated by small circles in FIG. 3, ofiers nearly the maximum possible to and the advantage of an analytical solution; for M 40 it is still not possible to trade (p for stiffness, but the optimum form is stiffer than the third-order form, the advantage increasing with increasing M.

Thus, the primary object of the invention is very satisfactorily achieved. A type of vibrator is disclosed having a value of (p( 1.6) about twice that of the stepped vibrator and about three-quarters of that of the exponential vibrator (2.0-2.4). This has been achieved in a fiexurally very stiff fomn, suitable for use as a tool or for numerous other purposes such as driving other vibrators. By way of example, for IOU-fold magnification the Fourier form is only 2.6 times less stiff than the stepped form, but 130 times stiffer than the exponential form.

Numerous and varied modifications and rearrangements of devices of the invention can be readily devised by those skilled in the art without departing from the spirit and scope of the principles of the invention. These principles are, inview of the disclosure of the present specification, broadly applicable to numerous arrangements in which mechanically vibratory members are employed.

What is claimed is:

1. A mechanical transformer for magnifying the amplitude of a repetitive, substantially sinusoidal, physical displacement by a factor M, M having a value exceeding 2, said transformer comprising an elongated acoustically resonant member having one end larger than the other end, a length 1 less than a wavelength for extensional Waves of the member's resonant frequency propagated along the member, a maximum transverse dimension not greater than one/half of said wavelength, the cross-sectional area A of the member varying between the larger and smaller ends of the member in accordance with the centimeters per second to said larger end, the smaller integral of the relation d(ln A) (4'y ('v -l'mzl u(9v (v 1) cos 1rX-(16-'y )a cos 1rX d(cos 77X) u(3-'Y (Y 42) TX+3F-('Y2 '1)COS2 ve4z C083 1rX end will have a particle velocity of 5000 centimeters per Where second.

As shown in- FIG. 6, and as described hereinabove, the corresponding exponentially tapered member would have the profile designated 80 and would obviously be extremely slender and lacking in stiffness for a major portion thereof toward the smaller end.

Contrasted with this, member 12" of FIG. 9 is still mechanically superior to member 12 of the Mason et al. patent from the standpoint of stiffness, notwithstanding the fact that member 12 provides ten times the magnification of member 12 of the patent.

To recapitulate, a p'rlmary object of the invention is to produce a type of amplitude transformer with greater ability to withstand large amplitudes (measured by the figure of merit, than that of the stepped transformer, but with only slightly lower flexural stiffness. The standard of comparison is the exponential transformer, the only other well known type of transformer, but which is very slender and hence of rather low stiffness at large magnification.

This object is achieved by deriving the family of profiles corresponding to displacement functions of thirdorder truncated Fourier series, Equation 11 with N=3. Owing to the special form of the result, these displacement functions essentially give a single useful solution 7 21 M 1 'Y-;,LL- M 1! where x represents the distance along the member from its larger end of the cross section for which the area A is being instantly determined and 0: is a constant at choice having a value between minus one and plus one.

2. The transformer of claim 1 in which 7 is chosen to give substantially the lowest ratio of maximum to minimum cross-sectional areas for the selected value of (x 3. The transformer of claim 2 in which 11 has a value of substantially 0.3.

4. The transformer of claim 1 in which ca has a value of substantially 0.3.

5. The transformer of claim 1 in which a is zero.

References Cited by the Examiner UNITED STATES PATENTS 2,017,695 10/35 Hahnemann 181-05 2,044,807 6/ 36 Noyes ISL-0.5 2,779,880 1/57 Malherbe 73--7l.5 X 2,948,154 8/60 Kleesattel.

BROUGHTON G. DURHAM, Primary Examiner. 

1. A MECHANICAL TRANSFORMER FOR MAGNIFYING THE AMPLITUDE OF A REPETITIVE, SUBSTANTIALLY SINUSOIDAL, PHYSICAL DISPLACEMENT BY A FACTOR M, M HAVING A VALVE EXCEEDING 2, SAID TRANSFORMER COMPRISING AN ELONGATED ACOUSTICALLY RESONANT MEMBER HAVING ONE END LARGER THAN THE OTHER END, A LENGTH 1 LESS THAN A WAVELENGTH $ FOR EXTENSIONAL WAVES OF THE MEMBER''S RESONANT FREQUENCY PROPAGATED ALONG THE MEMBER, A MAXIMUM TRANSVERSE DIMENSION NOT GREATER THAN ONE-HALF OF SAID WAVELENGTH, THE CROSS-SECTIONAL AREA A OF THE MEMBER VARYING BETWEEN THE LARGER AND SMALLER ENDS OF THE MEMBER IN ACCORDANCE WITH THE INTEGRAL OF THE RELATION 